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Energy methods for abstract nonlinear Schrödinger equations. (English) Zbl 1283.35128
Summary: So far there seems to be no abstract formulations for nonlinear Schrödinger equations (NLS). In some sense T. Cazenave [Courant Lect. Notes Math. 10, 323 p. (2003; Zbl 1055.35003), Chapter 3] has given a guiding principle to replace the free Schrödinger group with the approximate identity of resolvents. In fact, he succeeded in separating the existence theory from the Strichartz estimates. This paper is a proposal to extend his guiding principle by using the square root of the resolvent. More precisely, the abstract theory here unifies the local existence of weak solutions to (NLS) with not only typical nonlinearities but also some critical cases. Moreover, the theory yields the improvement of [N. Okazawa et al., Appl. Anal. 91, No. 8, 1605–1629 (2012; Zbl 1246.35189)].

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
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